Method and apparatus for computed tomography of portions of a body plane

ABSTRACT

A method and apparatus for computed axial tomography (CAT) image reconstruction applicable to X-ray scanning of the human body. Successive series of calculations determine the values of a characteristic in defined segmented areas of an examination plane. A reduction in the time and equipment requirements for reconstruction calculations may thereby be effected. 
     Images representative of the difference between the value of the characteristic at a reconstruction point and the average value of the characteristic in adjacent regions (Δμ) may be calculated and displayed from measurements taken within a localized contiguous region of the examination plane. Radiation dose to patients and computation time in X-ray computerized axial tomography scanning systems is thus reduced. Differential displays of the type described may be adjusted to image boundaries, in which case they do not suffer from gray scale resolution problems which are typical of prior art displays. 
     Image artifacts attributable to interpolation errors may be reduced with negligable effect on resolution by incorporating a weighting function, preferentially a Gaussian function, in the image reconstruction.

RELATED APPLICATIONS

This is a division of application Ser. No. 850,892, filed Nov. 15, 1977,now U.S. Pat. No. 4,433,380 which was a continuation-in-part of U.S.patent application Ser. No. 635,165 filed Nov. 25, 1975 and nowabandoned.

BACKGROUND OF THE INVENTION

1. Field of the Invention

The general field of this invention is tomography, that field thatrelates to obtaining an image of internal body parts in a plane throughthe body. Specifically, the field of this invention, called transverseaxial tomography, relates to a method and apparatus for calculating thevalue of an incoherent physical characteristic at points within a bodyfrom the values of a plurality of line integrals of the characteristictaken through the body. The invention is particularly suited forapplying a plurality of X-ray or gamma ray beams through a plane of abody, measuring the attenuation of each beam as it passes through thebody, and using the measurement information obtained to constructindividual attenuation coefficients for each element of a definedelement matrix in the body plane.

2. Description of the Prior Art

A prior art method and apparatus for transverse axial tomography isdescribed in U.S. Pat. No. 3,778,614 issued Dec. 1, 1973. That patentdescribes a technique to reconstruct a cross-sectional view of a bodyfrom a series of transmission measurements obtained by translating aradiation source and detector across the body section and repeating thistranslation motion at a number of angular orientations in the plane ofthe section. U.S. Pat. No. 3,778,614 is incorporated herein, byreference, as background material.

The objective of these measurements is to obtain, after computeranalysis of thousands of pieces of raw information about beamattenuation through the body plane, the attenuation coefficientassociated with each element of a matrix defined in the body plane. Themethod is useful for internal description of any body, but is primarilyuseful for identification of internal human body abnormalities. Theattenuation coefficients are different for normal body tissue, tumors,fat, etc. and consequently provide identifying information about softtissues in a human body. Especially useful for identification of braindisease and abnormalities, tomography by computer reconstructioneliminates obvious disadvantages of patient discomfort and morbiditynormally associated with brain investigations using pneumography,angiography and radioactive isotope scanning.

In one prior art method, the scan signals are processed to yield visualinformation and local values of the beam attenuation coefficients overthe body section. Detector scan signals are applied to an analog/digitalconverter to convert the analog scan signals which are proportional toeach beam attenuation to digital form and subsequently are recorded in astorage unit. Computer analysis of the entire matrix of scan signals,typically about 28,000 points, yields attenuation coefficientsassociated with a element matrix defined for the body. These attenuationcoefficients are related to the local physical properties in the bodyplane. After they are computed, the attenuation coefficients arerecorded in a storage unit, and subsequently converted to analog signalsby means of a digital/analog converter. These signals drive a viewingunit, typically a CRT, with the information content to pictoriallydisplay the attenuation coefficient for each matrix element. A permanentrecord of the display is achieved by means of a camera.

Another prior art method for tomographic image reconstruction makes useof a convolution and backprojection algorithm; as described, forexample, in U.S. Pat. No. 3,924,129 which is incorporated herein, byreference, as background material.

High spatial frequency components associated with X-ray scanningmeasurements can contribute to the production of artifacts inreconstructed images. U.S. Pat. No. 4,,002,911 describes a tomographicscanner wherein the intensity profile of an X-ray beam is weighted tolimit high frequency components. Weighting of the X-ray beam intensityprofile is mathematically equivalent to the inclusion of a weightingfunction in the convolution integral but is necessarily limited byphysical constraints of the X-ray system (i.e. weighting of an X-raybeam is limited to positive weighting functions).

At the present time substantial interest exists in the use of themethods and apparatus of computed tomography for the reconstruction ofX-ray images of the human body. The computational methods utilized are,however, equally applicable to other fields wherein the values of anincoherent characteristic function must be calculated from measurmentsof its line integrals. As used herein, the term "incoherentcharacteristic" means that the value of the characteristic at each pointaffects the value of the integral in a manner which is uncorrelated tothe value of the characteristic at other points. Although the method andapparatus for the invention are described herein with respect to X-rayscanning apparatus, the use of the invention is by no means limited tothat field.

A region of interest, to be examined by the methods of computedtomography, often occupies only a small fraction of the area of a planeextending through the body. For example, a radiologist may only beinterested in determining abnormalities within a single body organ. Adisadvantage of the prior art method and apparatus is that an entirebody plane must be scanned before local values of the attenuationcoefficients in a region of interest can be calculated. This is due tothe fact that attenuation of the X-ray beams access along the entirebeam path and affects the computation of attenuation coefficients atevery point in the plane. Thus, if a limited region of interest weretotally scanned and the surrounding body areas were only partiallyscanned (to the extent they were included in the scanning of the regionof interest) features in the partially scanned region would produceimage artifacts which would significantly distort computed values withinthe region of interest. Furthermore, severe restriction is placed on thestability of prior art X-ray tube and detector systems and upon themechanical precision of the scanning devices since consistent data mustgenerally be obtained over the entire scan time in order to accuratelycompute local attenuation coefficient values. Problems of reconstructionmay similarly arise in regions of the body which are subject to motionduring a scan.

A scanning motion consisting of translation followed by separaterotation is usually clumsy and subject to mechanical vibration and wear.Because of the mechanical problems involved, it is often difficult tospeed the sequence of translation and rotation movement to reducescanning time. Further problems are reated to the complexity of priorart computer programs necessary for reconstruction and thesophistication of the programs that are required.

SUMMARY OF THE INVENTION

In one embodiment of the invention a thin cross-section or plane througha body is examined by passing X- or gamma ray beams through a bodyplane. The body plane is depicted for examination purposes as atwo-dimensional matrix of elements defined by a plurality of concentriccircles which create concentric rings. The outermost ring is denoted asthe R ring, the next inner ring to the outermost ring described as theR-1 ring, and so on. Elements in the rings are created by dividing eachof the rings. In this manner, the notation N_(R) represents any numberequal to two or greater equally angularly spaced elements of theoutermost R ring, the R-1 ring being divided into N_(R-1) elements, andso on.

The method of determining individual attenuation or coefficients foreach element in the defined element matrix begins by rotating X- orgamma ray beams around the outside of the body, where a beam is providedfor each concentric ring and is so directed in the plane underinvestigation as to be continuously tangent to its associated ring.

From each beam emerging from the body, at N_(r) discrete angularintervals during the beams' rotation, a discrete output signal isrecorded representing the sum of the attenuation of the elements in eachrespective concentric ring intersected by the respective beam.

For the outmost R ring, the N_(R) discrete output signals from the beamtangent to the R ring are used in deriving signals proportional to theindividual absorption or transmission elements associated with each ofthe N_(R) elements in the R ring.

In response to the N_(R-1) discrete output signals from the beam tangentto the R-1 ring and the signals proportional to the individualattenuation coefficients from the beam tangent to the R-1 ring and thesignals proportional to the individual attenuation coefficientsassociated with the elements in the R ring through which the beamtangent to the R-1 ring passes at each of the N_(R-1) discrete angularintervals, signals are derived proportional to the individualattenuation coefficients associated with each of the N_(R-1) elements inthe R-1 ring.

This method is repeated for each succeeding ring in turn for ring R-2toward the center of the concentric circles. For each concentric ring,signals proportional to the individual attenuation coefficientsassociated with each of the elements in the ring are derived in responseto the N_(r) discrete output signals from the beam tangent to that ringand the previously derived signals proportional to the individualabsorption or transmission coefficients associated with the elements inall other rings through which the beam passes at each of the N_(r)discrete angular intervals.

The value of the attenuation coefficient at the center of rotation maythus be derived from equations which, in form, resemble the knowngeneral convolution-backprojection algorithm, but which include filter(weighting) functions which are derived from the concentric ring model.The computed radial origin of the ring model may, effectively, beshifted to permit calculation of the attenuation coefficient at anypoint in the plane.

Filter (weighting) functions utilized in the above-describedcomputational method have positive values for measurements passingthrough the origin of the computational system and negative values formeasurements in the adjoining ring. A numerical reconstruction of animage necessarily involves interpolation between these positive andnegative values, which interpolation may introduce significant imageartifacts. The introduction of an additional weighting function,preferably a Gaussian function, can substantially reduce interpolationerrors and errors associated with the propagation of statistical noisethrough the image without materially affecting image resolution.

The reconstructed value of the attenuation coefficient calculated at anyimage point is substantially influenced by the actual value of theattenuation coefficient at all other points in the plane. The effect ofthe attenuation coefficient at a remote point upon the calculated valueof an attenuation coefficient at a reconstruction point is substantiallya function of the reciprocal of the distance between the points.Generally, clinical requirements dictate that all points in a body planebe utilized for the reconstruction of attenuation coefficients at anyother point in the plane.

However if, instead of reconstructing the actual values of theattentuation coefficient at any point, the reconstruction insteadcalculates a differential image (Δμ) from values which represent thedifference between the value of the local attenuation coefficient at areconstruction point and the average value of the attenuationcoefficient in a surrounding area (a process which roughly correspondsto the calculation of the second spatial derivative of the localattenuation coefficient) the effect of the attenuation coefficient atremote points in the plane becomes dependent upon the reciprocal squareof the distance separating the remote points from the reconstructionpoint. The area scanned to produce Δμ images of a region of interestwithin the body may thus be confined to the region of interest and asmall surrounding transition area, the dimensions of the transition areabeing determined by the required precision of the reconstruction at theedge of the region of interest. Since X-ray radiation for suchmeasurements may, thereby, be confined to a small region of the body;the total radiation dose received by the patient and the associated scantime may be significantly reduced.

The methods of computed tomography generally yield images having a farwider range of values than it is possible to display on a conventionalimage output device. A radiologist examining computed tomography imagesmust, therefore, often shift the range and threshold of a gray-scaledisplay to allow imaging of significant clinical features. A detaileddiscussion of the nature of the gray-scale resolution problem and aprior art device for overcoming that problem is described in U.S. Pat.No. 4,030,119 which is incorporated herein, by reference, as backgroundmaterial.

Δμ displays, which display the difference between the local value of anattenuation coefficient and the average value of attentuationcoefficients in surrounding areas as signed, gray-scale levels (forexample with neutral gray representing zero; black representing negativevalues; and white representing positive values) tend to accentuateboundary regions, and thus do not suffer from the gray scale compressionproblems of prior art displays of the attenuation coefficient. Areaboundaries may thus be readily visualized in a Δμ gray-scale displaywhich might otherwise be invisible in a conventional gray-scale displayof attenuation coefficients.

The Δμ image affords the following properties:

The Δμ image provides the local departure of tissue properties from anaverage value of the surrounding tissues;

The Δμ image can provide the outline of tissue anomalies;

The scanning can be confined to a limited region of the body region withno significant distortion of the Δμ image reconstruction within the sameregion;

In a Δμ image which provides the outline of body organs and anomaliesthe effect of statistical noise is less important than in a conventionalimage. The Δμ image is used only to locate the boundaries rather than todetermine the local value of Δμ;

The rapid convergence of the filter functions which determine thebackprojection for the reconstruction of Δμ reduces the number ofsignificant terms of the series and the time required for thereconstruction calculations.

A novel apparatus is disclosed for performing the method. A rotatingframe is provided supported with respect to a fixed frame by means of abearing and is rotated therein. A source of X- or gamma rays is mountedon a first arm attached to the rotating frame. The source generates oneor more beams in a plane perpendicular to the axis of rotation. Thebeams are intercepted by a system of detectors mounted on a second armattached to the rotary frame. The beams are defined by collimatorsassociated with both the ray sources and the detection system anddirected so as to be effectively tangent to concentric rings definedabout the axis of rotation of the rotating frame in a plane of a bodyplaced in or near the axis of rotation between the source and detectorsystem.

In an embodiment of the detector system a reference crystal detector anda plurality of measurement crystal detectors are provided in groups,which may be moved in position on a track so as to intercept differentbeams passing through the body on different rotations of the rotatingframe. Photomultiplier tubes are provided, one for each measurementcrystal detector, to generate electrical signals proportional to thecorresponding beam intensity. Means are provided to magnetically storethe beam attenuation signals in digital form. A stored program digitalcomputer is provided for deriving signals proportional to attenuation ortransmission coefficients for the defined element matrix in the bodyplane. These signals are stored, and are then useful to provide arepresentation of the absorption characteristics of the body plane.

BRIEF DESCRIPTION OF THE DRAWINGS

This invention, as well as its objects and features, will be betterunderstood by reference to the following detailed descriptions of thepreferred embodiments of this invention taken in conjunction with theaccompanying drawings in which:

FIG. 1 is an X- or gamma ray source/detector orientation, constructed inaccordance with this invention, for rotation of a beam pattern about abody in which an element matrix is defined by concentric circles andequally spaced radii;

FIG. 2 shows in more detail the defined element matrix, constructed inaccordance with this invention, for measurement of absorptioncoefficients in a body plane;

FIG. 3 is a perspective of physical apparatus, constructed in accordancewith this invention, for rotating a beam pattern through a plane of abody and the measurement of beam attenuations after the beams passthrough it;

FIG. 4 is an X-ray tube beam spread, constructed in accordance with thisinvention, as it rotates about the body under investigation;

FIG. 5 is a schematic diagram of beam generation and detection inaccordance with this invention;

FIG. 6 is a schematic diagram of an alternative embodiment of beamgeneration and detection in accordance with this invention;

FIG. 7 is a schematic diagram of measurement data collection, recordingand processing in accordance with this invention;

FIG. 8 schematically describes a general scanning procedure;

FIG. 9 shows a change of coordinates between scanning and reconstructionframes of references;

FIG. 10 is a plotting of the values of F_(j) ;

FIG. 11 is a plotting of the values of G_(j) ;

FIG. 12 defines a scan transition region;

FIG. 13 is a poltting of the value of μ-<μ> across a plane interfacebetween uniform media;

FIG. 14 is a partial scanning procedure;

FIG. 15 illustrates the geometry of artifacts generated in a partialscanning approach;

FIG. 16 is a plotting of the values of μ and μ-<μ> within a partial scanregion;

FIG. 17 illustrates the generation of interpolation artifacts duringbackprojection;

FIG. 18 shows the effect of a Gaussian weighting function on the spatialresolution of a reconstruction of a uniform cylinder;

FIG. 19 is a reconstruction of the values of μ in a region of interestwhich was calculated from data obtained from a full scan of a bodyplane;

FIGS. 20-23 are reconstructions of Δμ values in a region of interestwhich were calculated from data scanned in a limited part of the bodyplane.

LIST OF PRINCIPAL SYMBOLS

In the following discussion the principal symbols are summarized anddefined as follows:

f: weighted attenuation value for μ reconstruction defined by Equation(2.22).

F_(j) : weighting coefficients to construct f; defined by Equation(2.23).

F_(j) : weighting coefficients to construct f; defined by Equation(3.2).

g: weighted attenuation value for (μ-<μ>) reconstruction; defined byEquation (2.28).

G_(j) : weighting coefficients to reconstruct g; defined by Equation(2.29).

G_(j) : weighting coefficients to reconstruct g; defined in Equation(3.6).

H_(j) : weighting coefficients for non-uniform radial samplingintervals; defined by Equation (6.21).

I: X-ray beam intensity

l: index of radius of averaging circle for <μ>.

N: number of angular sampling increments

r: radial coordinate of polar (r, θ) system in the image plane.

r₁ : radial increment

x: coordinate of Cartesian (x,y) system.

y: coordinate of Cartesian (x,y) system.

α_(i) : geometrical factor relating interception of beam in rings.

β: X-ray beam attenuation.

<β>: intensity-averaged measured value of β

Γ_(j) : weighting coefficients; defined by Equation (6.11).

δ, Δ: incremental difference operators.

θ: angular coordinate of polar (r, θ) system in the image plane.

θ_(i) : angular increment.

θ_(i),j : parametric coefficients; defined by Equation (2.6).

λ: generalized averaging dimension; defined by Equation (6:13).

μ: local value of linear attenuation coefficients.

<μ>: average value of μ over a circle of radius l_(r).spsb.1 about eachreconstruction point.

ξ: distance along a ray path from source to detector

ρ: radial coordinate of polar (ρ, 7/8) system in the the scanning plane.

ψ: angular coordinate of polar (ρ, ψ) system in the scanning plane.

ψ₁ : angular increment.

ψ₀ : angular sector scanned in an incomplete scanning.

ω: weighting function; defined by Equation (6.1).

DESCRIPTION OF THE PREFERRED EMBODIMENTS Concentric Ring Scanning

FIG. 1 shows a sketch of a body plane 111 to be examined by transverseaxial tomography according to this invention. The body 111 is assumed tobe placed between a source 300 of X- or gamma rays and a detector 301,which may be a scintillator and a photomultiplier and which preferablyalso includes a collimator. For illustrative purposes, detector 301 isassumed to be movable on a track 302 such that beams may be detectedwhich pass at various angles from the source through body 111. Multipledetectors, each with an associated collimator can of course be providedas detectors 301, 301', 301", etc., or multiple detectors may be movableon track 302. The X-ray source 300 and detectors 301, are attached to arotating ring 303 which is rotatable about an axis O perpendicular tothe body plane 111. Body 111 is shown in FIG. 1 coexistent with axis O,but it may be placed anywhere within the beam range of source 300 anddetector 301.

As shown in FIG. 1 a series of concentric circles is defined about axisof rotation O. As ring 303 rotates about the axis of rotation O, theX-ray beam or beams is continuously directed (as shown at oneorientation angle of rotation) perpendicular to subsequent radii fromaxis O at point P at all times as ring 303 rotates about axis O. As aresult, a beam such as 310 is at all times tangent to the outer ringabout center O as the source-detector system rotates.

FIG. 2 shows in more detail the concentric system defined about axis ofrotation O. Beam 310 is shown at a particular orientation during itsrotation about body 111 and is perpendicular to a particular radiusvector r at point P. By appropriate collimation, the beam width w can bemade to approximate the concentric ring width r₁. The example depictedin FIG. 2 shows beam 310 passing through the outermost concentric ring(i). Perpendicular to radius vector r, beam 310 is depicted as passingthrough elements labelled t=n_(i) -1, n_(i), 1, 2 and 3. These elementsare among those elements in the i ring, totalling n_(i) elements.

In order to describe the interior of body 111 according to the matrix ofelements throughout the concentric ring-radius vector system shown inFIG. 2, each small element is assigned an unknown value of attenuationcoefficient. For example, the attenuation coefficient for element t=1 inthe i th ring is designated μ_(i),l ; for element t=2, μ_(i),2 ; for thet th element, μ_(i),t. The measured beam attenuation for beam 310 shownwill be given by the sum of the average value of the linear attenuationconstants μ for each element through which the beam passes.

During rotation about axis O, the beam attenuation between source 300and detector 301 is obtained at n_(i) different positions, only one ofwhich is shown in FIG. 2. Beam attenuation for each measurement,designated β_(i),z is simply the sum of the linear attenuation constantsfor each element through which the beam passes, multiplied by anindividual geometrical factor determined by the interception of thebeams with each cell. The rotation-measurements steps of the beam 310,as source 300 and detector 301 rotate about O, are identified by anindex z. This index z runs from 1 in steps of 1 until z=n_(i), equal tothe number of elements in ring i. Thus, the measurement of the beamattenuation at each position of the first intercepting ring leads to theequations, ##EQU1## where Z=1, 2, . . . n_(i)

The term α_(i), t-z represents the geometrical factor determined by theinterception of the beam 310 with each element t as it rotates in zsteps about ring i.

Since t is taken equal to z, that is, the number of elements in ring iis t, and the number of measurements around ring i is equal to z,equation (1.1) represents a system of equations z=n_(i) in number,having t=n_(i) unknown parameters μ_(i),t. The solution of the system ofequations (1.1) yields the values of μ associated with each element onthe i=1 ring.

In the next scanning ring, the i-1 ring, the measurement of the beamattenuation leads to the new system of equations, ##EQU2## for, z=1,2, .. . n_(i-1)

wherein α'₁, t-z is the geometrical factor determined by theinterception of the beam (e.g. beam 311, FIG. 1) in the new ring, i-1,with the elements of the outer ring i.

The values μ_(i),t have been determined by the solution of Equations(1.1); the solution of the system of Equations (1.2) provides the valuesof μ_(i-1), t in the ring i-1. The measurement in each scanning ringwith decreasing radii provides a system of equations similar to (1.2)with terms on the right hand side containing known values of μ in theelements pertaining to the outer rings. It is apparent that the numberof elements of each outer ring which contributes to the attenuationalong an inner ring decreases rapidly as the scanning radius approacheszero, i.e. as the scanning beam approaches the center of rotation.

Thus, the local properties are fully determined upon completion of eachscanning ring without having to wait for the total scanning of the bodysection.

The number of equations in each set, similar to equation (1.2), isrelatively small and can be arranged to decrease as the interior ringswith smaller radii are measured. Assuming for example a scanning radiusof the outer ring of the order of 150 mm and an element width on theorder of 3 mm, each independent equation set for the outer ringsconsists of only several hundred equations. The solution for the unknownμ's for each ring sequentially from the outside ring toward the insiderings, requires far less computational time than prior art X-raytomographic systems. As the inner rings are measured, it is possible todecrease the number of measurements taken around the ring (i.e. definen_(i) to be less for the inner rings than for the outer rings, therebykeeping the element size approximately constant) with the result thatthe equation set size is reduced. Computational time is correspondinglyreduced for solution of inner ring μ's.

Concentric Ring Scanner

Illustrated in FIG. 3 is a perspective drawing of a concentric ringscanning apparatus. A fixed frame 600 supports a rotating frame 601which is free to revolve about an axis of rotation 602. A motor drive614 is provided in fixed frame 600 to propel rotating frame 601.Attached to rotating frame 601 are two arms 603, 604 spacedapproximately 180 degrees from one another. Arm 603 supports an X-raytube 605 and an associated X-ray tube collimator control 606. Arm 604carries a detector assembly 607 and associated detector collimators.

A couch 608 is provided to allow a human body 111 to be positionedbetween X-ray tube 605/-x-ray tube collimator control 606 and detectorassembly 607. Couch 608 is supported by couch support 609. A couchcontrol system 610 is provided which translates the couch 608 parallelto the axis of rotation 602, thereby positioning body 111 to a pointwhere beams from X-ray tube 605 may intersect a desired plane throughthe body 111. In addition, the couch control system 610 translates thecouch 608 in any direction in a plane perpendicular to the axis ofrotation, thereby positioning the axis of rotation close to the desiredarea of the body 111.

Since the X-ray tube 606 is rotatable about center line 602 means areprovided to cool it and provide it with high voltage electrical powerwhile it is rotating. These means, shown in modular form, are a coolingwater rotating assembly 611 and a high voltage slip ring assembly 612.Means must also be provided to send command and control signals to X-raytube 605 and its associated collimator assembly and collimatorsassociated with detectors 607 while they are rotating. Command andcontrol slip ring assembly 614 is provided for that purpose. Likewisedata transmission slip ring assembly 613 is provided to provide a meansfor transmission of data signals from detectors 607 while they arerotating.

FIG. 4 shows a preferred orientation of X-ray tube 656 and itsassociated collimator control 606 with respect to detector and detectorcollimator apparatus 604.

The X-ray beam 700 produced by the source is fan shaped and subtends anapical angle α. The tangents to the circle of the aperture opening 701which pass through the source define an apical angle β. The relationshipbetween the apical angle α and the apical angle β is that the apical βangle α is less than the apical angle β.

As indicated in FIG. 3, X-ray tube 656 and detector assembly 604 arerigidly connected to each other by arms 603, and 604 on rotating frame601. Rotation of the frame 601 about center line 602 (point 0 to FIG. 4)causes the X-ray beam 700 to sweep out a fan-shaped pattern, whichsubstantially covers any body placed within an aperture 701. Thescanning proceeds with sequential rotations of the X-ray beams, with atleast one beam being directed to at least one particular ring. On eachsubsequent rotation around the outside of the body the beam isredirected to at least one particular additional ring. In a preferredembodiment, the fan shaped beam subtends an approximately 30 degree arcas the X-ray tube-detector assemblies are rotated at speeds of up to onecomplete rotation per second for approximately 10 revolutions. Theaperture opening 701 is approximately 26 inches in diameter. The arms603, 604 attaching the X-ray housing 605 and detector system 607 areapproximately thirty inches long. The rotating frame 601 is supportedwith respect to fixed frame 600 by a single, thirty-five inch diameter,precision ball bearing.

FIG. 5 illustrates the multiple beam scanning aspects of this invention.The X-ray tube 605 emits a continuous fan-shaped array of X-rays, butthis continuous array must be divided into beams in order for themethods described previously in this specification to be applicable.Collimators 606 and 800 are provided to create a plurality of beamspassing through a cross section of a body 111 placed within aperture701. For illustrative purposes three detector system pairs consisting ofcrystal scintillators and photomultipliers (811, 820; 812, 821; 813,822) are shown in position 1. A reference scintillator 810 and itsassociated photomultiplier 823 are stationary. The detector pairs remainin position 1 for the first rotation of rotating frame 601 (FIG. 3). Atthe start of the second rotation, the detector system pairs are shiftedalong track 302 to position 2 for detection of beams at that position.The detectors are shifted to position 3 at the start of the thirdrevolution, and so on. This shifting of detectors at the end of onerotation and the beginning of another rotation assures that the entirebody placed within aperture 701 may be scanned.

FIG. 4a illustrates a method for scanning a body using a source ofpenetrating radiation that produces a plurality of coplanar beamsdefining a fan beam by:

(a) positioning the fan beam relation to a spacially fixed center at afirst position;

(b) rotation the fan beam about the fixed center and measuring theintensity of each beam after it passes through the cross section;

(c) repositioning the fan beam relative to the fixed center to a secondposition; and

(d) repeating step (b).

The rotations described in step (b) may be through substantially 360° sothat each rotation causes the fan beam to sweep out an area of the crosssection defined by reference circles concentric with the fixed centerand tangent to the extremities of the fan mean. The positions of the fanbeam may be selected such that the swept out areas are continuous.

In practice the translatory motion of the X-ray source and/or detectorsneed not be accomplished in stepwise fashion after each rotation.Rather, the translatory and rotational motions may be accomplishedsimultaneously so that the point of tangency of each X-ray beam moves ina smooth spiral. The data thus obtained may then be interpolated, forexample by a linear interpolation in a digital computer, to calculateequivalent data at points in the concentric ring coordinate system.

A preferred embodiment of the scanning system of FIG. 5 consists of anarrangement capable of scanning a test object contained within a 20 inchdiameter circle about axis of rotation O. Thirteen detector units areprovided one of which is the reference pair 810, 823, the other twelveof which are moveable to ten positions along detector track 302. Eachdetector system is used to scan a 2.5 degree sector of the totalscanning area, ten revolutions of the X-ray tube/detector system 604being used to scan the entire body.

Detector 810 and photomultiplier 823 are used to generate a referencebeam attenuation signal for all the other detectors to account for anyvariations with time in beam strength eminating from X-ray tube 605. Asshown in FIG. 5 a particular beam 855 is collimated by tube collimator606 and passes through an attenuator 850 located outside the location ofthe body being examined. The absorption characteristics of attenuator850 are preferably selected to be similar to that of the body beingexamined. Tissue equivalent plastic is an example of an attenuatormaterial suitable for this purpose. Detector pair 810,823 generates asignal, the intensity of which is proportional to the strength of theX-ray beam after absorption by attenuator 850.

Each detector pair for the beams passing through the body underinvestigation generates a signal proportional to a particular beam'sintensity after it passes through the body. The crystal scintillatorsproduce a high-frequency signal (visible light spectrum) proportional tothe number of photons in the X- or gamma ray beams impinging on them.The photomultipler tubes associated with each crystal scintillator,react to the light energy from their respective scintillators togenerate an electrical signal proportional to beam strength impinging onthe scintillators. For example, an electrical signal proportional to thebeam strength of beam 856 is generated at the output of photomultipliertube 820. Similarly, crystal scintillator/photomultiplier pairs generateoutput signals proportional to the strength of other beams at position1, position 2, etc. for the entire beam pattern after successiverotations of system 604.

In a preferred embodiment of this invention, the X-rays generated byX-ray tube 605 are collimated by means of a 15 cm long collimator 606 atthe X-ray tube source, and a 20 cm long collimator 800 at the detectorsystem 604. This collimation at the X-ray source and detector definesradiation beams having a rectangular profile of 1 mm by 5 mm width asmeasured by scanning a lead edge at the mid-point of the beam path.

The range of values for which the photomultiplier must respond can bereduced by covering the body being examined with a material, theabsorption of which is known, so that beam intensities received by thedetectors are kept as constant as possible as they pass through thebody.

FIG. 6 shows an alternate embodiment of detector orientation. Detectors910 and 911 are located on track 901, and detectors 920 and 921 arelocated on track 902. As shown, detectors 910 and 911 measure beamattenuation through circular rings, defined about rotation axis O,different from those measured by detectors 920 and 921. Multiplepositions on each track can be established and the detectors shifted inposition with each rotation until a defined ring matrix is entirelyscanned and detected. Collimators 606 are provided at the X-ray sourceand collimators 930 at the detectors are also provided.

An X-ray tube appropriate for the particular embodiment discussed aboveis a modified version of a Philips 160 kV Beryllium Window Tube, ModelMCN 160.

Appropriate detectors include scintillation detectors such as NaI, CaF₂, BGO and proportional counters such as high pressure xenon detectorsand solid state detectors.

FIG. 7 indicates how the beam attenuation data measured by the detectorsystems, including the photomultipliers 1000₁ 1000₂, . . . 1000₃, areprocessed during the rotational scanning of a body. An informationsignal is generated in each photomultiplier at each defined incrementfor each rotation of the X-ray source/detector system. These signals areindividually amplified by amplifiers 10101, 10102, . . . 10103, are eachtaken up in turn by serializer 1020, converted to digital form by analogto digital converter 1030, and stored in a data storage medium 1040 suchas magnetic tape, disk, or drum or solid state memory. This datacollection process continues for each detector position, for eachdefined increment step, for the complete rotation. During or after thedata collection process, a computer 1050 under direction of a storedreconstruction code program, processes the collected data according tothe methods discussed elsewhere in this specification. The output of thecomputer 1050 is a sequence of digital signals proportional to theattenuation coefficients of each element in the defined circular ringmatrix. These signals are stored in a data storage unit 106 which may beidentical to unit 1040 or similar to it. The output digital signals canthen be printed and/or converted to analog form and used to drive adisplay on a cathode-ray tube, thereby pictorially indicating theattenuation coefficients for the defined matrix in the cross section ofthe body being investigated.

GENERAL RECONSTRUCTION METHOD

The concentric ring reconstruction may be generalized for imagereconstruction at any point. In the plane x, y of FIG. 8, thecross-section of a body is confined within the boundary S. Line βrepresents the axis of an X-ray beam, which ideally is assumed to be ofnegligible cross section. The total attenuation of the beam passingthrough the body is given by ##EQU3## where I_(o) and I_(e) are the beamintensity at the entrance and exit of the body section respectively. Thetotal attenuation βcan be written in terms of the local value of theattenuation coefficient μ as the line integral ##EQU4## Assume now thatat each point P of the body section the values of β are available forany line ξ passing through P in any angular direction. From these valuesof β it is possible to compute the value of μ at each point of the bodysection. O_(r) is an arbitrary point where μ is to be computed. Define afamily of circles with center O_(r) and radii

    r.sub.j =jr.sub.1 (j=1, 1, 2, . . . )                      (2.3)

where r₁ is an arbitrary dimension which is small compared to the bodysection dimensions. Thus, the body section is divided into a largenumber of circular sectors.

At a point P_(j) over the circle of radius r_(j) let β_(j) be themeasured attenuation along a line tangent to the circle as indicated inFIG. 8.

Define μ_(h) as the average value of μ in the region between the circleof radius hr₁ and the circle of radius (h+1) r₁. Thus, μ_(o) is theaverage value of μ within the first circle of radius r₁. By averagingover 2π the values of β_(o) measured with the X-ray beams passingthrough O_(r) one has ##EQU5##

Similarly the average over 2π of the values of β_(j) is related to theaverage values of μ_(h) (for h ≧j by the equation ##EQU6## where##EQU7## from Equation (2.4), with the aid of Equation (2.5), oneobtains ##EQU8## where ##EQU9## with ##EQU10## The asymptotic value ofcoefficients K_(j) is ##EQU11## The numerical values of K_(j) up toj=100, are presented in Table I.

                  TABLE I                                                         ______________________________________                                                  j K.sub.j              j K.sub.j n                                  j K.sub.j l j        j  K.sub.j  l j                                          ______________________________________                                         1 .57735E+00                                                                           .57735E+00  51 .12502E-01                                                                            .98763E+00                                    2 .32826E+00                                                                           .74148E+00  52 .12261E-01                                                                            .98787E+00                                    3 .22153E+00                                                                           .81532E+00  53 .12029E-01                                                                            .98809E+00                                    4 .16542E+00                                                                           .85668E+00  54 .11806E-01                                                                            .98831E+00                                    5 .13153E+00                                                                           .88298E+00  55 .11591E-01                                                                            .98852E+00                                    6 .10903E+00                                                                           .90116E+00  56 .11384E-01                                                                            .98873E+00                                    7 .93064E-01                                                                           .91445E+00  57 .11183E-01                                                                            .98892E+00                                    8 .81161E-01                                                                           .92460E+00  58 .10990E-01                                                                            .98911E+00                                    9 .71953E-01                                                                           .93259E+00  59 .10804E-01                                                                            .98930E+00                                   10 .64621E-01                                                                           .93905E+00  60 .10623E-01                                                                            .98947E+00                                   11 .58645E-01                                                                           .94438E+00  61 .10449E-01                                                                            .98964E+00                                   12 .53681E-01                                                                           .94886E+00  62 .10280E-01                                                                            .98981E+00                                   13 .49493E-01                                                                           .95267E+00  63 .10117E-01                                                                            .98997E+00                                   14 .45912E-01                                                                           .95594E+00  64 .99583E-02                                                                            .99013E+00                                   15 .42814E-01                                                                           .95880E+00  65 .98048E-02                                                                            .99028E+00                                   16 .40109E-01                                                                           .96131E+00  66 .96560E-02                                                                            .99042E+00                                   17 .37725E-01                                                                           .96352E+00  67 .95117E-02                                                                            .99056E+00                                   18 .35610E-01                                                                           .96550E+00  68 .93716E-02                                                                            .99070E+00                                   19 .33719E-01                                                                           .96728E+00  69 .92356E-02                                                                            .99084E+00                                   20 .32019E-01                                                                           .96888E+00  70 .91035E-02                                                                            .99097E+00                                   21 .30482E-01                                                                           .97033E+00  71 .89751E-02                                                                            .99109E+00                                   22 .29087E-01                                                                           .97165E+00  72 .88502E-02                                                                            .99122E+00                                   23 .27813E-01                                                                           .97286E+00  73 .87288E-02                                                                            .99134E+00                                   24 .26647E-01                                                                           .97397E+00  74 .86107E-02                                                                            .99145E+00                                   25 .25575E-01                                                                           .97499E+00  75 .84958E-02                                                                            .99157E+00                                   26 .24585E-01                                                                           .97594E+00  76 .83838E-02                                                                            .99168E+00                                   27 .23670E-01                                                                           .97682E+00  77 .82748E-02                                                                            .99178E+00                                   28 .22820E-01                                                                           .97763E+00  78 .81686E-02                                                                            .99189E+00                                   29 .22029E-01                                                                           .97839E+00  79 .80651E-02                                                                            .99199E+00                                   30 .21291E-01                                                                           .97919E+00  80 .79641E-02                                                                            .99209E+00                                   31 .20601E-01                                                                           .97977E+00  81 .78657E-02                                                                            .99219E+00                                   32 .19955E-01                                                                           .98039E+00  82 .77697E-02                                                                            .99228E+00                                   33 .19348E-01                                                                           .98098E+00  83 .76760E-02                                                                            .99237E+00                                   34 .18776E-01                                                                           .98153E+00  84 .75845E-02                                                                            .99246E+00                                   35 .18238E-01                                                                           .98205E+00  85 .74952E-02                                                                            .99255E+00                                   36 .17729E-01                                                                           .98254E+00  86 .74079E-02                                                                            .99264E+00                                   37 .17248E-01                                                                           .98301E+00  87 .73227E-02                                                                            .99272E+00                                   38 .16793E-01                                                                           .98345E+00  88 .72394E-02                                                                            .99280E+00                                   39 .16361E-01                                                                           .98387E+00  89 .71579E-02                                                                            .99289E+00                                   40 .15951E-01                                                                           .98427E+00  90 .70783E-02                                                                            .99296E+00                                   41 .15560E-01                                                                           .98465E+00  91 .70005E-02                                                                            .99304E+00                                   42 .15189E-01                                                                           .98501E+00  91 .69243E-02                                                                            .99312E+00                                   43 .14834E-01                                                                           .98535E+00  93 .68498E-02                                                                            .99319E+00                                   44 .14496E-01                                                                           .98568E+00  94 .67768E-02                                                                            .99326E+00                                   45 .14173E-01                                                                           .98600E+00  95 .67054E-02                                                                            .99333E+00                                   46 .13864E-01                                                                           .98630E+00  96 .66355E-02                                                                            .99340E+00                                   47 .13569E-01                                                                           .98659E+00  97 .65671E-02                                                                            .99347E+00                                   48 .13285E-01                                                                           .98687E+00  98 .65000E-02                                                                            .99354E+00                                   49 .13014E-01                                                                           .98713E+00  99 .64343E-02                                                                            .99360E+00                                   50 .12753E-01                                                                           .98739E+00 100 .63699E-02                                                                            .99366E+00                                   ______________________________________                                    

The immediate question which arises in examining Equation (2.7) is howmany terms of the sum have to be included in the computation of thevalue of μ_(o) in order to perform the reconstruction within a givenerror. In other words, how far from O_(r) has the body section to bescanned over the sequence of circular orbits to reconstruct thedistribution of μ in a limited region around O_(r) ? The answer to thisquestion largely depends upon the dynamic range of values of β_(j).Assume, for example, that the distribution of values of β_(j) is not farfrom uniform (as it would be approximately in the case of a water filledcompensation bag enclosing the body under scrutiny). The contribution ofthe terms outside of the range j=j would be ##EQU12## As a consequenceof this slow rate of decrease with j it is apparent that, with thevalues of r₁ in the millimeter range, both scanning and computation mustinclude almost the entire body section if the error in thereconstruction has to be maintained within a small limit, for example,1%.

PARTIAL SCANNING

The result expressed by Equation (2.11) is equivalent to saying that thecontribution of the scanning of an area of the body section located at adistance r from O_(r) affects the computation of μ_(o) as r⁻¹. Thus, ifone computes the differences of the values of μ at two points close toeach other, the scanning of a surrounding area affects the difference ofμ as

    1/r.sub.1 -1/r.sub.2                                       (2.12)

where r₁, r₂ are the distance of the area from the two points. Thus, forlarge values of r₁, r₂ the scanning of an area of the body sectionaffects the differences of values of μ essentially as r₁,2⁻². By using adifferential-like method in the image reconstruction, it is possible toconfine both scanning and computation to a limited area of the bodysection. As one example of a mathematical analysis of this method,consider first the average value of μ within a circle of radius lr. Theaverage <μ> is given by ##EQU13## By virtue of Equation (2.5), Equation(2.13) transforms to ##EQU14## where ##EQU15## The coefficients K_(j) inEquation (2.7) and KHD o,j in Equation (2.14) satisfy the asymptoticcondition ##EQU16## Thus, from Equations (2.7) and (2.14) one obtains##EQU17## where ##EQU18## Asymptotically the coefficient φ_(e),jdecreases as j⁻⁴ and this rapid rate of decay is the basis for alocalized scanning and image reconstruction.

An important property of both equations (2.7) and (2.17) is the uniformaveraging process of the attenuation measurements over each circle ofthe image reconstruction sequence, as a result of the integration over2π. Thus, the effect of the statistical fluctuations of the individualmeasurements of β is minimized uniformly over the entire reconstructionarea.

Both Equations (2.7) and (2.17) provide the solution of thereconstruction problem. The reconstruction point O_(r) is an arbitrarypoint in the x, y plane and Equations (2.7) and (2.17) assume that thevalues of the attenuation data β have been measured over the family ofcircles concentric with O_(r). Assume that the attenuation measurementshave been conducted in the polar system of coordinates ρ, ψ of FIG. 9 insuch a way that the values of β are known over the family of circlesconcentric with the origin O. From Equation (2.7), the value of μ at apoint O_(r) of polar coordinates r,θ is given by ##EQU19## FIG. 9 showsthe locus of the points of tangence of the circles of center O with theX-ray beams which are located at a distance r_(j) =jr₁ from thereconstruction point O_(r). From Equation (2.17) the value of μ-<μ> atO_(r) is ##EQU20## where ##EQU21## Both equations (2.19) and (2.20) canbe written in the same form of the reconstruction solution obtained witha convolution approach. Consider the function ##EQU22## where thecoefficients F_(j) are ##EQU23## Coefficients F_(j) are the weightingfunctions and they satisfy the condition ##EQU24## In Equation (b 2.22)one has

    β](h-j) r.sub.1 ψ]=β[|(h-j) r.sub.1 |, ψ+π]                                               (2.25)

when

    h-j<0                                                      (2.26)

Equation (2.19) transforms to ##EQU25## The coefficients F_(j) areplotted in FIG. 10 and the numerical values of F_(j) are given in theTable II for J ≦100.

                  TABLE II                                                        ______________________________________                                        j      -F.sub.j       j      -F.sub.j                                         ______________________________________                                         1     .57735E+00     51     .24514E-03                                        2     .16413E+00     52     .23579E-03                                        3     .73842E-01     53     .22697E-03                                        4     .41355E-01     54     .21863E-03                                        5     .26306E-01     55     .21075E-03                                        6     .18172E-01     56     .20328E-03                                        7     .13295E-01     57     .19620E-03                                        8     .10145E-01     58     .18949E-03                                        9     .79948E-02     59     .18311E-03                                       10     .64621E-02     60     .17706E-03                                       11     .53313E-02     61     .17129E-03                                       12     .44734E-02     62     .16581E-03                                       13     .38071E-02     63     .16058E-03                                       14     .32794E-02     64     .15560E-03                                       15     .28543E-02     65     .15084E-03                                       16     .25068E-02     66     .14630E-03                                       17     .22191E-02     67     .14197E-03                                       18     .19783E-02     68     .13782E-03                                       19     .17747E-02     69     .13385E-03                                       20     .16009E-02     70     .13005E-03                                       21     .14515E-02     71     .12641E-03                                       22     .13221E-02     72     .12292E-03                                       23     .12093E-02     73     .11957E-03                                       24     .11103E-02     74     .11636E-03                                       25     .10230E-02     75     .11328E-03                                       26     .94559E-03     76     .11031E-03                                       27     .87665E-03     77     .10747E-03                                       28     .81500E-03     78     .10473E-03                                       29     .75692E-03     79     .10209E-03                                       30     .70971E-03     80     .99552E-04                                       31     .66456E-03     81     .97107E-04                                       32     .62359E-03     82     .94752E-04                                       33     .58629E-03     83     .92481E-04                                       34     .55224E-03     84     .90291E-04                                       35     .52108E-03     85     .88178E-04                                       36     .49248E-03     86     .86138E-04                                       37     .46617E-03     87     .84169E-04                                       38     .44192E-03     88     .82266E-04                                       39     .41951E-03     89     .80426E-04                                       40     .39876E-03     90     .78648E-04                                       41     .37952E-03     91     .76928E-04                                       42     .36164E-03     92     .75264E-04                                       43     .34499E-03     93     .73654E-04                                       44     .32946E-03     94     .72094E-04                                       45     .31496E-03     95     .70584E-04                                       46     .30140E-03     96     .69120E-04                                       47     .28870E-03     97     .67702E-04                                       48     .27678E-03     98     .66327E-04                                       49     .26558E-03     99     .64993E-04                                       50     .25505E-03     100    .63699E-04                                       ______________________________________                                    

In a similar manner Equation (2.20) can be written again with thedefinition of the function ##EQU26## where ##EQU27## and the followingcondition is satisfied ##EQU28## A plotting of G_(j) is shown in FIG. 11and the values of G_(j) are given in Table III for j<100 in theparticular case of l=11. One observes the inversion of sign of the termsG_(j) for i>l.

                  TABLE III                                                       ______________________________________                                        j      +G.sub.j       j      +G.sub.j                                         ______________________________________                                         1     -.58689E+00    51     .86350E-05                                        2     -.17424E+00    52     .79768E-05                                        3     -.84126E-01    53     .73804E-05                                        4     -.51720E-01    54     .68389E-05                                        5     -.36715E-01    55     .63463E-05                                        6     -.28608E-01    56     .58975E-05                                        7     -.23748E-01    57     .54877E-05                                        8     -.20611E-01    58     .51130E-05                                        9     -.18470E-01    59     .47699E-05                                       10     -.16944E-01    60     .44551E-05                                       11     .23816E-01     61     .41660E-05                                       12     .94539E-02     62     .39000E-05                                       13     .50385E-02     63     .36549E-05                                       14     .30838E-02     64     .34288E-05                                       15     .20470E-02     65     .32200E-05                                       16     .14342E-02     66     .30269E-05                                       17     .10449E-02     67     .28481E-05                                       18     .78441E-03     68     .26823E-05                                       19     .60305E-03     69     .25284E-05                                       20     .47279E-03     70     .23855E-05                                       21     .37682E-03     71     .22525E-05                                       22     .30460E-03     72     .21286E-05                                       23     .24925E-03     73     .20132E-05                                       24     .20616E-03     74     .19055E-05                                       25     .17216E-03     75     .18050E-05                                       26     .14500E-03     76     .17110E-05                                       27     .12307E-03     77     .16230E-05                                       28     .10519E-03     78     .15406E-05                                       29     .90485E-04     79     .14634E-05                                       30     .78291E-04     80     .13910E-05                                       31     .68105E-04     81     .13230E-05                                       32     .54541E-04     82     .12591E-05                                       33     .52293E-04     83     .11991E-05                                       34     .46126E-04     84     .11426E-05                                       35     .40849E-04     85     .10893E-05                                       36     .36311E-04     86     .10392E-05                                       37     .32391E-04     87     .99189E-06                                       38     .28990E-04     88     .94726E-06                                       39     .26027E-04     89     .90511E-06                                       40     .23435E-04     90     .86528E-06                                       41     .21160E-04     91     .82762E-06                                       42     .19155E-04     92     .79200E-06                                       43     .17384E-04     93     .75827E-06                                       44     .15814E-04     94     .72631E-06                                       45     .14419E-04     95     .69603E-06                                       46     .13174E-04     96     .66731E-06                                       47     .12062E-04     97     .64006E-06                                       48     .11065E-04     98     .61418E-06                                       49     .10169E-04     99     .58961E-06                                       50     .96627E-05     100    .56625E-06                                       ______________________________________                                    

In a way similar to Equation (2.11) one can compute in Equation (2.28)the order of magnitude of the contribution of the terms of the sumoutside of j=j with the assumption of a quasi-uniform distribution ofvalues of β_(j). One has ##EQU29## Equation (2.17) can be written againin the form ##EQU30## By virtue of Equations (2.18), Equation (2.32)reduces to Equation (2.27) in the limit l →√=. Thus Equation (2.32) maybe considered a more general solution of the reconstruction

Equation (2.32) defines the approach of the localized scanning. Due tothe rapidly diminishing value of the contribution of the terms of thesecond sum in Equation (2.17), a progressivelv larger error in themeasurement of β can be tolerated for increasing values of j (i.e. forincreasing distance from the region of reconstruction of the Δμimage).Beyond a given distance the measurement of β becomes unnecessary andassumed β values can be substituted for the actual measured data withoutintroducing a significant error in the calculation of Δμ.

In the limit of radius of l approaching unity, the reconstructed valueΔμ acquires the property of the local average of the second derivativeof μ. This is illustrated in FIG. 3 which shows the value of Δμ as afunction of the distance from a plane interface M between two uniformmedia. The value of Δμ is zero at the interface and at a large distancefrom the interface. Finite values of Δμ are confined to the region + or-lr₁ from the interface.

Thus the relationship between a Δμ image and a μ image is controlled bythe relation of the parameter l.

If one has to extract the local value of μ from the Δμ image anindependent knowledge or measurement of the value of μ within eachradius l r₁ is required. This average may be is known beforehand or itmay be obtained from a total scanning of the body section. However, thelatter need only have a low spatial resolution if l is large compared tounity and, as a consequence, the total scanning has less stringentrequirements on the stability of the attenuation measurements ascompared to a high spatial resolution scanning.

Two categories of clinical intentions can be identified where, inprincipal, the Δμ image is of diagnostic value per se.

If the images are used to diagnose localized density perturbations inessentially uniform areas (the liver is an example) and the value of lis such that the perturbations fit well within the averaging circle, theinformation contained in the Δμ image provides the full diagnosis of theanomaly or departure from usual tissue properties. A scanning localizedto the area of interest provides the set of β measurements required forthe Δμ image. The precision of the reconstruction is placed in thedifference between local and normal values μ rather than the absolutevalues of μ.

The second category corresponds to images of the interface of bodyorgans and boundaries of tissue anomolies. These images correspond tosmall values of l, for example l=2, where the boundaries are identifiedby a distribution of positive and negative values of Δμ as shown in FIG.13 (as long as the radius of curvature of the boundary is larger thanseveral pixal sizes).

EFFECTS OF SCANNING PROCEDURE ON IMAGE RECONSTRUCTION

To compute either function g(hr₁,ψ) or f(hr₁,ψ), it is necessary toextract the values of β from the measurements of the X-ray beamintensity according to Equation (2.1). In an actual scanning andreconstruction procedure the values of I_(o) and I_(e) in Equation 2(1)are measured at the entrance and exit of the body section respectively.At a first glance it would then appear that the reconstructioncalculation is dependent upon the values of intensity I_(o) outside ofthe body. However, it is easy to write solutions of μ and μ-<μ> in aform which shows that a knowledge of I_(o) is not required. To do so,write f(hr₁, ψ) in the form ##EQU31## where ##EQU32## By virtue of 2.24,F_(j) satisfies the condition ##EQU33## Consequently in Equation (3.1),the value of f does not change is the values of β are changed into a newdistribution β' such that

    β'[(h+j)r.sub.1, ψ]=β[(h+j)r.sub.1, ψ]+β.sub.o (ψ)                                                   (3.4)

where β_(o) is an arbitrary constant independent of j. In a similar wayg(hr₁, ψ) can be written as ##EQU34## where ##EQU35## G_(j) alsosatisfies the condition ##EQU36## Thus g (hr₁, ψ) is also independent ofan additive arbitrary constant β_(o) in the values of the attenuationmeasurements.

ARTIFACTS GENERATED BY PARTIAL SCANNING

In the case of a partial scanning, the attenuation data are collectedonly within a circle of radius r_(s), resulting in an error in thereconstruction of either μ or μ-<μ>. This error is essentially due tothe superposition of artifacts generated by the incomplete scanning ofpoints located outside of the circle of radius r_(s).

Assume a δ-like object located at a point P at a distance r_(p) >r_(s)as shown in FIG. 14 and reconstruct the value of either μ or μ-<μ>within the circle of radius r_(s) with no other object in the scanningplane.

If the distance of each scanning ray from the center 0 never exceedr_(s), the attenuation data are collected between the angles ψ, ψ+ψ_(o)formed by the two lines which are perpendicular to the tangents a₁, a₂from P to the circle of radius r_(s) as shown in FIG. 14. Thus point Pis scanned only within the fraction ψ_(o) /π of the total scanningcycle, regardless of the scanning procedure. Accordingly the presence ofthe δ-object outside of the scanning circle generates an artifactdistribution which is essentially oriented along the two lines a₁ anda₂. In the limit of r_(p) →r_(s), i.e., for p approaching the pointP_(s) on the scanning circle, one has ψ_(o) →π and the two lines a₁, a₂coverage in a single line at P_(s) tangent to the circle.

As a consequence the partial scanning introduces an error in the imagereconstruction within the circle of radius r_(s), which dependsprimarily upon the distance of the reconstruction point from the linesa₁, a₂ rather than the distance from the location P of the object leftoutside of the scanning circle. Hence the maximum error is found closeto the points of tangence of a₁, a₂ with the circle.

The reconstructed value of μ within a region of the scanning circleclose to either a₁, or a₂, is described by an approximate solutionwritten in the form: ##EQU37## where r is the distance of thereconstruction point from P and j₁, j₂ are given by ##EQU38## η beingthe angle between a₁ and the line PO_(r) as shown in FIG. 15. The valueof μ-<μ> at the same point is obtained simply by substituting μ-<μ> to μand G_(j) to F_(j) in Equation (4.1), i.e. ##EQU39## The maximum valuesof both μ and μ-<μ> are found within a distance from a₁, a₂ of the orderof r₁, and the order of magnitude of the maxima are ##EQU40##

Thus the two maxima are of the same order of magnitude regardless of thevalue of the parameter l, and decrease slowly with the distance of Pfrom the points of tangence. As indicated by the shaded regions of FIG.15, only a small fraction of the circle area, close to the periphery ofthe scanning circle is affected by the maximum amplitude of theartifacts given by Equations (4.4).

As the distance from either a₁ or a₂ increases, the reconstructed valueof μ decreases rather rapidly. For values of η close to 0 and ψ₀ and inthe intervals ##EQU41## the value μ is ##EQU42## where

    α.sub.1 =η; α.sub.2 =ψ.sub.o -η    (4.7)

μ-<μ> decreases also and attains a negative value at values of η whoseorder of magnitude is given by ##EQU43## and for values of η ##EQU44##the magnitude of μ-<μ> decreases very rapidly with either α₁ or α₂according to the equation ##EQU45## i.e., the magnitude of μ-<μ>decreases inversely to the third power of the distance from either a₁a₂. As a consequence, within a distance lr₁ from the periphery of thecircle the Δμ image reconstruction is virtually unaffected by thepresence of the δ-object outside of the scanning circle. A plotting ofboth μ and μ-<μ> within the angular interval ψ_(o) is shown in FIG. 16.

RECONSTRUCTION COMPUTATION

The image reconstruction will, in general, be implemented numerically ina general purpose digital computer which may, additionally, includededicated array processing hardware. The specific solutions are, ofcourse, highly dependent on such factors as the required computationalspeed, accuracy, and capital investment. The following discussions is,therefore, intended to enable those skilled in the computer programmingart to effectively implement numerical solutions without undueexperimentation.

The general solution to the reconstruction problem is stated by Equation(2.32), since in the limit l→∞, <μ>→0 and G_(j) →F_(j) which are theweighting functions for reconstruction of μ. Numerical implementation ofthe reconstruction algorithm can be discussed in terms of Equations(2.28), (2.29) and (2.32), recognizing that Equations (2.22), (2.23) and(2.27) are recovered as a special case. Therefore, the differencesbetween a partial scanning and reconstruction and a total scanning andreconstruction reside in the magnitude of the requirements forinformation storage, reconstruction speed, and interpretation of thesolution, but not in the form of the equations or the logic of theinstructions to implement their solution. Accordingly, a singlereconstruction code may be developed in which specification of the valuel is the only parameter which distinguishes a "partial reconstruction",producing the solution for μ-<μ>, from a "total reconstruction",producing the solution for μ.

The reconstruction code may be structured in four basic modules. Thefirst module calculates and stores the weighting functions (Equations2.23 and 2.29); ##EQU46## and an auxiliary function; ##EQU47## Selectionof the value of j_(max) must be made on the basis of accuracy andcomputational speed; these considerations will be elaborated in thefollowing discussion.

The second module of the reconstruction code calculates the function for g (depending of the value of l) in accord with Equations (2.22) or(2.28): ##EQU48## where the scanning is carried out over a maximumradius r_(s) =j_(s) r₁ and the reconstruction is to be carried outwithin a circle of radius r_(o) =j_(o) r. The values of β(r, ψ) may beobtained from direct measurement as provided in prior art scanninginstrumentation. The functions β₀,1 (ψ) and β₀,1 (ψ) represent either(a) the background attenuation measured outside the body in the case ofa total scanning i.e., β₀,1 =β₀,2 =β₀ (ψ), or (b) suitableapproximations to the attenuation in the compensation region (see FIG.12) in the case of a partial scanning, e.g., β₀,1 =β(r_(s), ψ) and β₀,2=β(r_(s), ψ+π). The transformation stated by Equations (2.25) and (2.26)should be recalled in connection with Equation (5.3).

If the values of β are reasonably uniform, the value of j_(max) requiredto achieve a specified degree of accuracy can be estimated form Equation(2.30) or (2.24). The error ε incurred by truncating the summation afterj_(max) terms is simply ##EQU49## Thus, ε→∞ however, as will bedemonstrated below, an "exact" solution, in the sense that ##EQU50## canbe achieved with a finite value of j_(max), dictated only by j_(s) andl.

It can be seen from Equation (5.3) that j_(max) =j_(s) +j_(o) isrequired to complete the indicated summation in this equation. Considerfirst the case of a total scanning of the body section; j_(o) =j_(s) isusual in this case. As pointed out above, an arbitrary constant β_(o)(ψ), may be added to or substracted from the measured attenuation valueswithout affecting the solution for f(r,ψ) (assuming j_(max) →∞).Therefore, the values of β(r,ψ) in Equation (5.3) may be normalized toβ(r,ψ)-β_(o) (ψ), yielding; ##EQU51## in which case the auxiliaryfunction γ_(j) is not needed, and j_(max) =2j_(s) is required tocomplete an "exact" solution. However, for typical values of j_(s)needed to carry out a total scan of a body section with a high degree ofprecision, the computational requirement to include 2j_(s) terms in thecalculation of f(fr₁, ψ) at each of 2j_(s) -1 radial positions, for eachangle ψ, may still present an unacceptable limitation on reconstructionspeed. On the other hand, use of j_(max) < introduces an error of order##EQU52## into the solution, which must be assessed in terms of therequired precision of the reconstruction.

In the case of a partial scanning, the reconstruction region should beconfined to a circle of radius r_(o) =r_(r) -lr₁ =(j_(s) -l)r₁. Thusj_(max) =2j_(s) =l is required to carry out the indicated summations inEquation (5.3). If the values of β(r,ψ) are normalized with respect to alinear function of r in this case,

    β.sub.n (r,ψ)=β.sub.o,1 +(β.sub.o,1 -β.sub.o,2)(r-r.sub.s)/2r.sub.s (-r.sub.s <r<r.sub.s) (5.5)

Equation (6.5) becomes: ##EQU53## which only requires j_(max) =2j_(s) -lterms for an "exact" solution and, again, deletes the auxiliary functionγ_(j).

Since the maximum scanning radius r_(s) =j_(s) r for a partial scanningis presumably much smaller than that required for a total scanning, therequired value of j_(max) for an "exact" solution is correspondinglyreduced. Therefore, calculation time for the function g, which isroughly proportional to j_(s) ², should not be a limiting factor inachievement of acceptable reconstruction speed in the case of a partialreconstruction.

It should be pointed out that since the auxiliary function γ_(j) may becomputed, and stored, for arbitrarily large values of j_(max) in thefirst module of the reconstruction code, at very modest computationalexpense, Equation (5.3) may be preferred over (5.4) or (5.6), sincenormalization of the data is not required. Equations (5.4) and (5.6)demonstrate that the minimum number of weighting terms required forexact total and partial reconstruction are 2j_(s) and 2j_(s) -lrespectively, if the attenuation values are normalized to β₀,1 =β₀,2 =0.

The third module of the reconstruction code is the "backprojection" orreconstruction step, per se, as given by Equation (2.32) or (2.27):##EQU54## wherein the transformation given by Equations (2.25) and(2.26) has been utilized. The integration may be carried out by thesimple trapezoidal rule procedure using equally spaced angularintervals, typically the same as the angular increment in the scanningdata. The required values of the integrand are linearly interpolatedfrom the calculated values.

The fourth, and final module of the reconstruction code is the imagedisplay. The reconstructed distribution of μ or Δμ over the scanningplane may be displayed by either (a) assigning a grey scale to the rangeof values of μ or (μ-<μ>) and a pixel size to each coordinate point (x,y)=(rcosθ, rsinθ) to produce a photographic type image, or (b) searchingthe distribution for the contour lines μ=constant or (μ-<μ>)=constantwhich can be plotted as continuous functions of (x,y) or (r,θ). Detailsof both techniques are well known in the art; however, it should bepointed out that generation of a grey scale image is relatively fast andqualitatively informative, whereas the contour line technique providesmore quantitative detail, both in terms of spatial and densityresolution, but at a much greater computational expense. Obviously,plots of μ or Δμ as a function of position along selected lines withinthe scanning plane may also be obtained in place of, or in addition to,the image display.

CORRECTION OF ARTIFACTS DUE TO FINITE SAMPLING

The reconstruction algorithm is based on the assumption that μ is acontinuous function of position which changes slowly over the elementalstep r₁. Because of the interpolation procedures, artifacts aregenerated in the presence of discontinuities, such as interfaces betweenregions with largely different values of μ. Errors and artifacts arealso generated by the finite radial and angular sampling intervals ofany scan procedure.

In FIG. 17 O_(r) is a reconstruction point. During the summation orbackprojection of data that numerically replaces the integration inEquation 2.27 some interpolation must be made between proximal values off, such as f[jr₁, ψ] and f[(j+1) r₁, ψ]. Interpolation artifacts will begenerated if f suffers a large change in this interval. Specifically thesign inversion of F_(j) and G_(j) from a positive value at j=0 tonegative values at j=±1 is responsible for the large interpolationerrors in the presence of discontinuities of the distribution of μ. Thereconstruction algorithm may be optimized by modifying the weightingfunctions in such a way as to minimize the interpolation error and theresulting reconstruction noise. A modification of the algorithm leadingto a reduction of the computational reconstruction noise will also havea beneficial effect on image artifacts generated by statistical noise inthe data acquisition system.

Obviously, any change in the shape of the weighting functions intendedto minimize the reconstruction noise must be analyzed in terms of itseffect on the reconstructed values of both μ and μ-<μ> and in particularon the spatial resolution of the reconstructed images. To discuss thisproblem it is convenient to modify the reconstruction approach discussedabove by substituting the calculation of μ at each point O_(r) (r,θ)with the calculation of a weighted average μ of the attenuationcoefficient as defined by the equation ##EQU55## where r, θ are thepolar coordinates relative to the reconstruction point O_(r) ##EQU56##and ω (r) is a continuous function of the distance r from O_(r), whichsatisfies the condition ##EQU57## Assume a family of circles concentricwith O_(r) and radii

    r.sub.h= hr.sub.1                                          (6.4)

Equation (8.1) can be written in the form ##EQU58## where μ_(h) is theaverage value of μ between the circles of center O_(r) and radii hr₁ and(h+1) r₁, and ##EQU59## In the manner described above, one obtains theaverage values μ_(h) ##EQU60## where the coefficients K_(j),h arerelated to the parameters θ_(j),k defined in Equation (2.6) by ##EQU61##Thus the value of μ given by Equation (6.5) can be written as ##EQU62##where ##EQU63## and ##EQU64## The relationship between the reconstructedvalue of μ and the actual value of the attenuation coefficient μ dependsupon the selection of function ω (r) which determines the parametersM_(j) in the coefficients Γ_(j). Assume for instance that it is aGaussian function ##EQU65## where the dimension r_(o) is related to r₁by

    r.sub.o =λr.sub.1                                   (6.13)

λ being an arbitrary positive number. With the particular function ωgiven by Equation (6.12), in a first approximation, the value of μmaintains the significance of an average value of the attenuationcoefficient within a circle of radius λr₁ and the coefficients M_(j)become ##EQU66##

                                      TABLE IV                                    __________________________________________________________________________    j =.250000                                                                              =.500000                                                                              =1.000000                                                                             =2.000000                                                                             =4.000000                                                                             =10.000000                          __________________________________________________________________________       .100000E+01                                                                           .981684E+00                                                                           .632121E+00                                                                           .221199E+00                                                                           .605869E-01                                                                           .995017E-02                         1                                                                              -.577350E+00                                                                          -.556201E+00                                                                          -.163134E+00                                                                           .109536E+00                                                                           .577497E-01                                                                           .111488E-01                         2                                                                              -.164130E+00                                                                          -.166309E+00                                                                          -.194570E+00                                                                          -.352457E±01                                                                        .380649E-01                                                                           .110394E-01                         3                                                                              -.738420E-01                                                                          -.742578E-01                                                                          -.841560E-01                                                                          -.775770E-01                                                                           .129448E-01                                                                           .100671E-01                         4                                                                              -.413549E-01                                                                          -.414827E-01                                                                          -.443852E-01                                                                          -.575726E-01                                                                          -.722496E-02                                                                           .865788E-02                         5                                                                              -.263065E-01                                                                          -.263575E-01                                                                          -.274887E-01                                                                          -.347911E-01                                                                          -.183476E-01                                                                           .699149E-02                         6                                                                              -.181722E-01                                                                          -.181963E-01                                                                          -.187244E-01                                                                          -.219224E-01                                                                          -.213030E-01                                                                           .520595E-02                         7                                                                              -.132949E-01                                                                          -.133077E-01                                                                          -.135862E-01                                                                          -.151109E-01                                                                          -.193324E-01                                                                           .342184E-02                         8                                                                              -.101451E-01                                                                          -.101525E-01                                                                          - .103130E-01                                                                         -.111357E-01                                                                          -.155668E-01                                                                           .174130E-02                         9                                                                              -.799477E-02                                                                          -.799934E-02                                                                          -.809827E-02                                                                          -.858510E-02                                                                          -.118946E-01                                                                           .243563E-03                        10                                                                              -.646205E-02                                                                          -.646503E-02                                                                          -.652929E-02                                                                          -.683713E-02                                                                          -.901663E-02                                                                          -.101796E-02                        11                                                                              -.533133E-02                                                                          -.533335E-02                                                                          -.537689E-02                                                                          -.558159E-02                                                                          -.696186E-02                                                                          -.201603E-02                        12                                                                              -.447342E-02                                                                          -.447484E-02                                                                          -.450539E-02                                                                          -.464702E-02                                                                          -.553060E-02                                                                          -.274725E-02                        13                                                                              -.380715E-02                                                                          -.380817E-02                                                                          -.383023E-02                                                                          -.393146E-02                                                                          -.451774E-02                                                                          -.322775E-02                        14                                                                              --.327940E-02                                                                         -.328016E-02                                                                          -.329649E-02                                                                          -.337081E-02                                                                          -.377651E-02                                                                          -.348773E-02                        15                                                                              -.285428E-02                                                                          -.285485E-02                                                                          -.286720E-02                                                                          -.292303E-02                                                                          -.321459E-02                                                                          -.356570E-02                        16                                                                              -.250679E-02                                                                          -.250724E-02                                                                          -.251674E-02                                                                          -.255952E-02                                                                          -.277564E-02                                                                          -.350326E-02                        17                                                                              -.221913E-02                                                                          -.221948E-02                                                                          -.222692E-02                                                                          -.226025E-02                                                                          -.242448E-02                                                                          -.334073E-02                        18                                                                              -.197831E-02                                                                          -.197858E-02                                                                          -.198449E-02                                                                          -.201085E-02                                                                          -.213822E-02                                                                          - .311418E-02                       19                                                                              -.177467E-02                                                                          -.177489E-02                                                                          -.177964E-02                                                                          -.180077E-02                                                                          -.190126E-02                                                                          -.285355E-02                        20                                                                              -.160094E-02                                                                          -.160112E-02                                                                          -.160498E-02                                                                          -.162212E-02                                                                          -.170256E-02                                                                          -.258206E-02                        21                                                                              -.145154E-02                                                                          -.145169E-02                                                                          -.145486E-02                                                                          -.146891E-02                                                                          -.153413E-02                                                                          -.231632E-02                        22                                                                              -.132212E-02                                                                          -.132224E-02                                                                          -.132487E-02                                                                          -.133650E-02                                                                          -.138998E-02                                                                          -.206715E-02                        23                                                                              -.120928E-02                                                                          -.120938E-02                                                                          -.121158E-02                                                                          -.122128E-02                                                                          -.126558E-02                                                                          -.184064E-02                        24                                                                              -.111029E-02                                                                          -.111038E-02                                                                          -.111223E-02                                                                          -.112039E-02                                                                          -.115741E-02                                                                          -.163929E-02                        25                                                                              -.102298E-02                                                                          -.102306E-02                                                                          -.102463E-02                                                                          -.103155E-02                                                                          -.106273E-02                                                                          -.146313E-02                        26                                                                              -.945586E-03                                                                          -.945649E-02                                                                          -.946990E-03                                                                          -.952892E-03                                                                          -.979364E-03                                                                          -.131063E-02                        27                                                                              -.876654E-03                                                                          -.876708E-03                                                                          -.877861E-03                                                                          -.882926E-03                                                                          -.905546E-03                                                                          -.117939E-02                        28                                                                              -.814996E-03                                                                          -.815042E-03                                                                          -.816038E-03                                                                          -.820411E-03                                                                          -.839959E-03                                                                          -.106671E-02                        29                                                                              -.759623E-03                                                                          -.759663E-03                                                                          - .760528E-03                                                                         -.764323E-03                                                                          -.781139E-03                                                                          -.969870E-03                        30                                                                              -.709709E-03                                                                          -.709744E-03                                                                          -.710499E-03                                                                          -.713808E-03                                                                          -.728426E-03                                                                          -.886378E-03                        __________________________________________________________________________     The resulting values of Γj are shown in Table IV for several values     of λ. In the limit of λ small compared to unity,     ##EQU67##     and Γ.sub.j reduces to the value of F.sub.j, as is apparent from     Table IV for λ=0.25. Conversely in the limit of λ large     compared to unity, Γ.sub.j is positive for j<λ, consistent     with the behavior of the coefficients K.sub.j, K.sub.o,j which determine     the average value of μ within the circle of radius λr.sub.1, in     Equation (2.14). Of particular interest is the smooth transition from     positive to negative values of Γ.sub.j for λ>>1, with a     minimum value of Γ.sub.j found at j larger than λ as shown in     Table IV. Thus a value of λ of the order, or smaller than unity,     leads to a value of μ close to the local value of μ as given by     Equation (2.27), without altering the spatial resolution of the     reconstructed image in any significant manner. An example of the effect of     changing the parameter λ Equation (6.14) is provided by the     reconstruction of the μ image of a uniform cylinder as shown in FIG.     18, (based on a computational simulation of attenuation data in a cylinder     coaxial with the axis of scanning). The cylinder radius is equal to     10r.sub.1 ; FIG. 18 shows the values of μ versus the radial distance     from the axis for values of λ equal to 0.25, 1, 1.5, 2, 4. The     reconstruction of μ across the boundary of the cylinder, at     r=10r.sub.1, shows the increasing loss of spatial resolution above     λ=1. Outside of the cylinder the reconstruction error fluctuates     about zero and Table V shows the effect of λ on the values of μ     within the radial interval 70<r/r.sub.1 >99.

                                      TABLE V                                     __________________________________________________________________________    R/R1   =0.25   =1.00   =1.50   =2.00   =4.00                                  __________________________________________________________________________    .99000E+02                                                                           -.74756E-03                                                                           -.217044E-03                                                                           .249783E-04                                                                           .315018E-04                                                                          -.12950E-04                            .98000E+02                                                                           -.37380E-02                                                                           -.178138E-02                                                                          -.677949E-03                                                                          -.367232E-03                                                                          -.81693E-04                            .97000E+02                                                                           -.69320E-02                                                                           -.358714E-02                                                                          -.150025E-02                                                                          -.763108E-03                                                                          -.13642E-03                            .96000E+02                                                                           -.10141E-01                                                                           -.540831E-02                                                                          -.233824E-02                                                                          -.117515E-02                                                                          -.21002E-03                            .95000E+02                                                                           -.93699E-02                                                                           -.492432E-02                                                                          -.205338E-02                                                                          -.100945E-02                                                                          -.18377E-03                            .94000E+02                                                                           -.66138E-02                                                                           -.328842E-02                                                                          -.120310E-02                                                                          -.547142E-03                                                                          -.12731E-03                            .93000E+02                                                                           -.39392E-02                                                                           -.174577E-02                                                                          -.457485E-03                                                                          -.157787E-03                                                                          -.72186E-04                            .92000E+02                                                                           -.13691E-02                                                                           -.324347E-03                                                                           .150706E-03                                                                           .140403E-03                                                                           .57641E-05                            .91000E+02                                                                            .11879E-02                                                                            .700432E-03                                                                           .313436E-03                                                                           .132251E-03                                                                           .17880E-04                            .90000E+02                                                                            .33729E-02                                                                            .158540E-02                                                                           .467702E-03                                                                           .143026E-03                                                                           .15827E-04                            .89000E+02                                                                             .45014E-02                                                                           .235809E-02                                                                           .908587E-03                                                                           .405469E-03                                                                           .97493E-04                            .88000E+02                                                                            .55441E-02                                                                            .303988E-02                                                                           .124678E-02                                                                           .556555E-03                                                                           .12360E-03                            .87000E+02                                                                            .65729E-02                                                                            .370779E-02                                                                           .157099E-02                                                                           .693512E-03                                                                           .15605E-03                            .86000E+02                                                                            .70248E-02                                                                            .399090E-02                                                                           .170359E-02                                                                           .743631E-03                                                                           .15229E-03                            .85000E+02                                                                            .31792E-02                                                                            .179735E-02                                                                           .776416E-03                                                                           .358307E-03                                                                           .97608E-04                            .84000E+02                                                                           -.69335E-03                                                                           -.423523E-03                                                                          -.178533E-03                                                                          -.555437E-04                                                                           .61978E-05                            .83000E+02                                                                           -.46344E-02                                                                           -.271753E-02                                                                          -.121274E-02                                                                          -.545647E-03                                                                          -.12926E-03                            .82000E+02                                                                           -.83592E-02                                                                           -.498473E-02                                                                          -.236067E-02                                                                          -.112102E-02                                                                          -.20656E-03                            .81000E+02                                                                           -.54654E-02                                                                           -.339375E-02                                                                          -.159221E-02                                                                          -.657532E-03                                                                          -.10503E-03                            .80000E+02                                                                           -.26544E-02                                                                           -.189733E-02                                                                          -.971910E-03                                                                          -.370553E-03                                                                          -.34637E-04                            .79000E+02                                                                            .28291E-04                                                                           -.489359E-03                                                                          -.510177E-03                                                                          -.292840E-03                                                                          -.39820E-04                            .78000E+02                                                                            .11563E-02                                                                            .770354E-03                                                                           .405253E-03                                                                           .185032E-03                                                                           .34493E- 04                           .77000E+02                                                                            .22317E-02                                                                            .197570E-02                                                                           .126268E-02                                                                           .598178E-03                                                                           .65952E-04                            .76000E+02                                                                            .29734E-02                                                                            .247217E-02                                                                           .158058E-02                                                                           .794099E-03                                                                           .97960E-04                            .75000E+02                                                                            .35697E-02                                                                            .252617E-02                                                                           .150811E-02                                                                           .785963E-03                                                                           .80164E-04                            .74000E+02                                                                            .39666E-02                                                                            .235006E-02                                                                           .117572E-02                                                                           .601905E-03                                                                           .85774E-04                            .73000E+02                                                                           -.10172E-03                                                                           -.223963E-03                                                                          -.216054E-03                                                                          -.122017E-03                                                                           .81227E-06                            .72000E+02                                                                           -.50516E-02                                                                           -.331927E-02                                                                          -.190677E-02                                                                          -.106602E-02                                                                          -.12221E-03                            .71000E+02                                                                           -.32284E-02                                                                           -.249043E-02                                                                          -.168228E-02                                                                          -.101827E-02                                                                          -.87943E-04                            .70000E+02                                                                           -.84907E-03                                                                           -.107589E-02                                                                          -.996119E-03                                                                          -.691295E-03                                                                          -.57945E-04                            __________________________________________________________________________

The of λ is particularly pronounced on the large error of μ of the orderof -10⁻² which is found in the proximity of r/r₁ =96 for λ=0.25. Table Vshows that the error at r/r₁ =96 decreases rapidly with increasingvalues of λ, and in particular the error is approximately halved forλ=1. Hence a substantial improvement of the reconstruction interpolationerror is achieved without a significant loss of spatial resolution.Equation (6.12) is only an example of a continuous function ω, whichyields an optimum form of the reconstruction algorithm as a trade-offbetween a value of μ sufficiently close to μ and a minimum amplitude ofthe computational noise.

In this formulation of the reconstruction algorithm, the differencebetween local value μ and average value <μ>, can be readily computedfrom Equation (6.9) as the difference between two values of μ for λ˜1and λ˜<1, i.e. ##EQU68## Asymptotically for j>>l, one has ##EQU69## andthe difference between the two values of Γ_(j) for λ˜1, λ˜l decreases asj⁻⁴. Thus solution (6.16) maintains the same properties of solution 2.23of the problem of localized scanning. In the numerical applications ofEquation (6.18), the reconstruction interpolation errors are generatedprimarily by the first term of the itegrand (λ˜1). Consequently, theartifacts generated in the numerical reconstruction procedure of theμ-<μ> image have essentially the same amplitude of the artifactsgenerated in the reconstruction of the local value of the attenuationcoefficient.

Equation (6.14) represents the solution of the direct problem ofcomputing the values Γ_(j) of the weighting function from a specifiedfunction of μ to be reconstructed in the image plane. The inverseproblem can be stated: if a particular shape of the weighting functionis specified, one may compute the function ω in Equation (6.11) anddetermine the relationship between the reconstructed value μ and theactual attenuation coefficient.

From Equation (6.11) one obtains: ##EQU70## Thus from Equation (6.6) theaverage value ω_(h) of ω between the circles of radii hr₁ and (h+1)r₁ is##EQU71## and finally, by means of Equation (6.19), Equation (6.6)provides the value of μ at each reconstruction point O_(r).

EXAMPLES OF A Δμ IMAGE RECONSTRUCTION

FIGS. 19-23 illustrate the transition from a μ to a Δμ image obtainedwith different values of the parameter l. FIG. 19 is a reconstruction ofμ values from data obtained in a Philips translational scanner(Tomoscan® 200 manufactured by Philips Medical Systems, Inc. Shelton,Conn. FIG. 19 is a partial reconstruction within a circular region whichincludes the liver, obtained with a full set of scanning data bothinside and outside of the circle. The radius of the circle, normalizedto r₁ l, is equal to 127.

FIGS. 20-23 are a set of Δμ images for values l=60, 40, 20 and 5obtained by ignoring scanning data outside of the circle of FIG. 19. Thegray scale of each image consists of sixteen equally spaced gray levels;the middle level corresponds to Δμ=0. Thus the images of FIGS. 20-23,present the full range of values of Δμ with the negative valuescorresponding to the darker half of the gray scale and the positivevalues corresponding to the lighter half. The images of FIGS. 14-23 areobtained by assuming that outside of the circle the value of β an eachradial line is constant and equal to the value measured on the circlefor that particular line. The difference between the l=60 (FIG. 20) andl=40, (FIG. 21) images is minor and both are close to the conventionalimage of values shown in FIG. 17. The overall range of Δμ values of thereconstructed image decreases with decreasing values of l and thedecrease in l results in a sharper transition across the interfaces asshown in the l=20 (FIG. 22) image. This becomes even more apparent inthe l=5 (FIG. 23) image which reduces to the outline of the body organswith values of μ-<μ> small everywhere else, (almost within the noiselevel); within the bone and in the soft tissue area as well.

It is worthwhile pointing out that the lack of actual scanning dataoutside of the reconstruction circle has a negligible effect on thereconstruction of the images of FIGS. 20 and 21 and the image distortionis confined to a very small annular close to the boundary of the circleeven for the larger values of l. This is the main reason why the Δμalgorithm allows a partial scanning of the area of interest.

The value of l represents an additional parameter in the display of areconstructed image which can be used, for example, to enhance thegeometry of interfaces in the area under scrutiny. In this connection itis worthwhile pointing out that for small values of l Equation (2.32)acquires the essential property of a local average of the secondderivative of μ. The value of μ-<μ> is zero at the interface between twouniform media. Thus in a general situation of image reconstructionacross sharp boundaries between media of different physical properties(like soft tissue-air interface of soft tissue-bone interface) theboundary would be described by one of the family of equations.

    μ-<μ>=0

provided that the radius of the averaging circle is smaller than thelocal radius of curvature of the interface in the scanning plane. HenceΔμ images provide a very convenient tool to outline either bone or softtissue interfaces without the need of computing the local values of theattenuation coefficient.

We claim:
 1. A method for scanning a body as part of a process forreconstructing a cross-section thereof using a source of penetratingradiation that produces a plurality of coplanar beams defining a fanbeam passing through the body coplanar with the cross-section, themethod comprising:(a) positioning the fan beam relative to a spatiallyfixed center at a first position; (b) rotating the fan beam about thefixed center and measuring the intensity of each beam after it passesthrough the cross-section; (c) repositioning the fan beam relative tothe fixed center to a second position; and (d) repeating step (b). 2.The method of claim 1 wherein the rotations are through substantially360° and each rotation causes the fan beam to sweep out an area of thecross-section defined by reference circles concentric with the fixedcenter and tangent to the extremities of the fan beam.
 3. The method ofclaim 2 wherein the positions of the fan beam are selected such that theswept out areas are contiguous.